Niven [3] gave a simple proof that π is irrational. Koksma [2] modified Niven's proof to show that er is irrational for every non-zero rational r. Dixon [1] made a similar modification to show that π is not algebraic of degree 2. In this note, we prove a general theorem which gives Niven's and Koksma's results as easy corollaries. A suitable modification in our proof also gives Dixon's result.

In this note we show that a solvable group of unipotent matrices over a skew field can be simultaneously triangularized.

It is shown that the classical Fourier transform can be extended as an algebra isomorphism onto the algebra of all complex-valued functions, which are measurable and finite a.e., under pointwise addition and multiplication. The extended Fourier transform agrees with the distributional Fourier transform on the space of all distributions which have regular transforms. It is defined on an algebra of Mikusiński-type operators in which multiplication is convolution in the subspace of integrable distributions.

Let C[0,1] be the space of all continuous real valued functions defined in [0,1] with the supremum norm

1

The subspace of C[0,1] consisting of all functions f(x) for which f(0) and f(l) are integers will be denoted by C0[0,1], Let be a sequence of real numbers satisfying:

2

A base a pseudoprime is an integer n such that

1

A Carmichael number is a composite integer n such that (1) is true for all a such that (a, n ) = l. It was shown by Carmichael [1] that, if n is a Carmichael number, then n is the product of k(>2) distinct primes P1,P2,P3, … Pk, and Pi-1|n-1(i=1, 2, 3, …, k).

A short proof is given of a biquadratic reciprocity law proved by Burde in 1969.

Let p and q be primes ≡1 (mod 4) such that (p | q) = (q | p) = 1. Then there are integers a, b, c, d with

Set